The role of connectivity in the theory of saturated/unsaturated flow through swelling media

A thermodynamically consistent model for multiphase flow which allows for connected and disconnected phases in a swelling medium is developed using hybrid mixture theory with three spatial scales. The mesoscale of the medium consists of swelling particles and two bulk phases, such as liquid water and vapor. The particles are a combination of a vicinal liquid and a solid, which may swell or shrink as a result of interaction with the other bulk phases; an example is a mixture of montmorillonite platelets and water. The theory defines connected and disconnected bulk phases of liquid and vapor at the mesoscale to create a dual-porosity type model at the macroscale. The disconnected vapor phase consists of either buoyant bubbles or confined vapor packets. The incorporation of disconnected and connected phases is useful for modeling unsaturated swelling systems. The macroscale solid phase volume fraction is refined from previous hybrid mixture approaches for two-phase multiscale problems and is fully utilized in the field equations and the constitutive theory. Macroscale equations for each of the six phases are presented with bulk regions separated into connected and disconnected domains. A constitutive theory is derived by exploiting the entropy inequality for the mixture. Generalized Darcy's laws and the final set of field equations for the system are presented and compared with previous hybrid mixture theoretic results. Classical parallel flow models only consider disconnected bulk regions and therefore are only appropriate for drainage; the current system can be useful for both imbibition and drainage, including extremely dry systems.